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Questions tagged [cayley-hamilton]

For questions about the Cayley-Hamilton theorem, which states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation.

If A is an $n×n$ matrix and $I_n$ is the $n×n$ identity matrix, the characteristic polynomial of A is defined as

$$ p(\lambda) = \det(\lambda I -A) $$

In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation; that is, $p(A)=0$. This is deeper than it appears at first: note that this does not follow by putting $\lambda=A$, as that is an abuse of notation.

There are several proofs; one is to approximate $A$ by diagonal matrices and then invoke continuity. Another uses Nakayama's Lemma.

Sources:

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Cauchy's integral formula for Cayley-Hamilton Theorem

I'm just working through Conway's book on complex analysis and I stumbled across this lovely exercise: Use Cauchy's Integral Formula to prove the Cayley-Hamilton Theorem: If $A$ is an $n \times n$ matrix over $\mathbb C$ and $f(z) = \det(z-A)$ is…
Sam
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Is this 1-line proof of Cayley–Hamilton incomplete?

In the comments of Martin Brandenburg's answer to this old MO question Victor Protsak offers the following "1-line proof" of the Cayley–Hamilton theorem. Here $p_A(\lambda)$ is the characteristic polynomial. Let $X = A - \lambda I_n$, then…
Qiaochu Yuan
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Interpreting the Cayley-Hamilton theorem

The statement of the Cayley-Hamilton Theorem is fairly straight-forward. I now know how to find characteristic polynomials from a given matrix (or at least a matrix with certain properties that I am unaware of!). I know that the eigenvalues of the…
The Chaz 2.0
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Is the proof of this lemma really necessary?

To prove the Cayley-Hamilton theorem in linear algebra, my professor said that a lemma was necessary: Lemma: Let $A \in M_n(\mathbb{K})$ be an $n\times n$ matrix over a field $\mathbb{K}$, let $b(t) \in M_n(\mathbb{K})[t]$ and $P(t) = b(t)[A-tI]$,…
Andy
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How many ways are there to prove Cayley-Hamilton Theorem?

I see many proofs for the Cayley-Hamilton Theorem in textbooks and net, so I want to know how many proofs are there for this important and applicable theorem?
user217174
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Is the converse of Cayley-Hamilton Theorem true?

The question is motivated from the following problem: Let $I\neq A\neq -I$, where $I$ is the identity matrix and $A$ is a real $2\times 2$ matrix. If $A=A^{-1}$, then the trace of $A$ is $$ (A) 2 \quad(B)1 \quad(C)0 \quad (D)-1 \quad…
user9464
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Cayley Hamilton Theorem Intuition

Why should, intuitive (not a formal proof, just motivations ) be true that the square matrix satisfy its own characteristic equation?
2569cfa
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Proof of Cayley-Hamilton Theorem in infinite fields only?

While trying to prove the Cayley-Hamilton theorem, I came up with the following proof: If $A$ is a diagonalizable matrix, so $A=SDS^{-1}$ with $D$ diagonal, then, letting $$P(\lambda)=\det(A-\lambda I)=\sum_{i=0}^n c_i\lambda^i,$$ $$``P(A)" =…
Carl Schildkraut
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Why is a matrix of indeterminates diagonalizable?

Fix $n^2$ indeterminates $t_1,\dots, t_{n^2}$. Let $A$ be the algebraic closure of $\mathbb C(t_1,\dots, t_{n^2})$. Consider the $n\times n$ matrix over $A$ whose entries are precisely $t_1, t_2,\dots, t_{n^2}$. Why is this diagonalizable? I feel I…
Potato
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Cayley-Hamilton theorem on square matrices

Can anyone help me by giving the proof of the Cayley-Hamilton theorem? It states that every square matrix $A$ satisfies its own characteristic equation: $$p_{A}(A) = 0$$ I could prove it when $A$ has distinct eigenvalues, because then it will have…
Goodarz Mehr
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Does the Cayley–Hamilton theorem work in the opposite direction?

The Cayley–Hamilton theorem states that every square matrix satisfies its own characteristic equation. But does it work in the opposite direction? If for example for a certain matrix $A$ we know that $ A^2-6A+9I=0, $ does that mean that the…
Ido
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Proofs of the Cayley-Hamilton Theorem

The idea of this post is for people to post different proofs of the Cayley-Hamilton Theorem. You can either try to post your own proof or give a reference. If you usse a reference, please give some ideas of what the proof looks like ;). I really…
Aitor Iribar Lopez
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Doubt over the proof of Cayley- Hamilton heorem

I am having some doubt in the proof of Cayley Hamilton theorem. This theorem says that every matrix is a root if its characteristic polynomial. Proof goes as follows: Let us assume that matrix $A$ is of order $n\times n$. If $P(\lambda)$ be its…
mathscrazy
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Show that $\exp(\alpha B)$ is represented by $I+(\sin\alpha)B+(1-\cos\alpha)B^2$ (Tokyo entrance exam [Math,2020])

I am currently trying to solve the following problem from the 2020 Tokyo entrance exam (math department): 第一問 正方行列A,Bおとすう。$$A=\begin{pmatrix}1&\sqrt2&0\\\sqrt2&1&\sqrt2\\0&\sqrt2&1\end{pmatrix}\quad…
CrSb0001
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Prove $\det((AB)^{n}-(BA)^{n})$ is a perfect cube.

We have $A,B$ two $3×3$ matrices with integer numbers. We know that $(AB)^{2}+BA=(BA)^2+AB$. a) Show that $\det((AB)^{n}-(BA)^{n})$ is divisible by $det(AB-BA)$. b) Show that if $\det(AB-BA)=1$, then $\det((AB)^{n}-(BA)^{n})$ is a perfect cube. I…
Stefan Solomon
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